Abstract. Topological order of a topological phase of matter in two spacial dimensions is encoded by a unitary modular (tensor) category (UMC). A group symmetry of the topological phase induces a group symmetry of its corresponding UMC. Gauging is a well-known theoretical tool to promote a global symmetry to a local gauge symmetry. We give a mathematical formulation of gauging in terms of higher category formalism. Roughly, given a UMC with a symmetry group G, gauging is a 2-step process: first extend the UMC to a G-crossed braided fusion category and then take the equivariantization of the resulting category. Gauging can tell whether or not two enriched topological phases of matter are different, and also provides a way to construct new UMCs out of old ones. We derive a formula for the H 4 -obstruction, prove some properties of gauging, and carry out gauging for two concrete examples.

We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin TQFTs at low energy. We formulate a 16-fold way conjecture for the minimal modular extensions of super-modular categories to spin modular categories, which is a categorical formulation of gauging the fermion parity. We investigate general properties of super-modular categories such as fermions in twisted Drinfeld doubles, Verlinde formulas for naive quotients, and explicit extensions of P SU (2) 4m+2 with an eye towards a classification of the low-rank cases.

We develop categorical and number theoretical tools for the classification of super-modular categories. We apply these tools to obtain a partial classification of super-modular categories of rank 8. In particular we find three distinct families of prime categories in rank 8 in contrast to the lower rank cases for which there is only one such family. 2 2. PRELIMINARIES In this section, we first introduce the notion of super-modular categories and some of its properties. Most of the results can be found in ([10, 13]) and the references therein. Then we discuss the Galois symmetry for super-modular categories.2.1. Centralizers. Whereas one may always define an S-matrix for any ribbon fusion category B, it may be degenerate. This failure of modularity is encoded it the subcategory of transparent objects called the Müger center B ′ . Here an object X is called transparent if all the double braidings with X are trivial:Generally, we have the following notion of the centralizer of the braiding.Definition 2.1. The Müger centralizer of a subcategory D of a pre-modular category B is the full fusion subcategoryWhile the notation D ′ is slightly ambiguous as it is relative to an ambient category, the context will always make it clear. By a theorem of Bruguières [8], the simple objects inSymmetric fusion categories have been classified by Deligne in terms of representations of supergroups [21]. In the case that B ′ ∼ = Rep(G) (i.e., B ′ is Tannakian), the de-equivariantization procedure of Bruguières [8] and Müger [36] yields a modular category B G of dimension dim(B)/|G|. Otherwise, by taking a maximal Tannakian subcategory Rep(G) ⊂ B ′ , the deequivariantization B G has Müger center (B G ) ′ ∼ = sVec, the symmetric fusion category of super-vector spaces. Generally, a braided fusion category B with B ′ ∼ = sVec as symmetric fusion categories is called slightly degenerate [22], while if B ′ ∼ = Vec, B is non-degenerate. The symmetric fusion category sVec has a unique spherical structure compatible with unitarity and has Sand T -matrices: S sVec = 1 √ 2 and T sVec = 1 0 0 −1 . Definition 2.4.Ŝ andT are called the Sand T -matrix of the fermionic quotient. By the following proposition, pointed super-modular categories always splits.4

We classify all modular categories of dimension 4m, where m is an odd square-free integer, and all ranks 6 and 7 weakly integral modular categories. This completes the classification of weakly integral modular categories through rank 7. Our results imply that all integral modular categories of rank at most 7 are pointed (that is, every simple object has dimension 1). All strictly weakly integral (weakly integral but non-integral) modular categories of ranks 6 and 7 have dimension 4m, with m an odd square free integer, so their classification is an application of our main result. The classification of rank 7 integral modular categories is facilitated by an analysis of actions on modular categories by two groups: the Galois group of the field generated by the entries of the S-matrix and the group of isomorphism classes of invertible simple objects. The interplay of these two actions is of independent interest, and we derive some valuable arithmetic consequences from their actions.

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